Concurrency Theory: Lecture 3, 21 August 2019
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Confusion

- Was a conflict resolved?  Occurrence of an independent
  transition creates/removes a conflict.

Conflict(t,M) = { t' | M--t--> and M--t'--> and t,t' in conflict}

Confusion occurs when M --t'-->M' and for some t independent of t',
Conflict(t,M) =/= Conflict(t,M').

- Conflict set can either grow or shrink (or both)

- Two fundamental types of confusion, symmetric and asymmetric

- Confusion reflects interaction between concurrency and conflict
  (choice) and is one of the reasons why analysis of concurrent
  systems is harder than sequential systems

Free choice nets

- Avoid confusion by restricting  net structure
- If Pre(t) and Pre(t') intersect, then Pre(t) = Pre(t').

- Can derive stronger decidability/algorithmic results for free
  choice nets: empirical evidence that lack of confusion
  simplifies analysis.

Deadlock and liveness

- M is deadlocked (or "dead") if no t is enabled at M.

- t is dead at M if t is not enabled in Reach(M)

- t is live at M if it is not dead at any marking in Reach(M)

    For all M' in Reach(M), there exists M" in Reach(M') such
    that M" --t-->

- A net is live at M if all transitions are live at M

  Many early results in Petri net theory focus on
  characterizations of live and safe initial markings for nets

Reachability

Is M in Reach(Min)?

- Decidable, but proof is hard: no bound on size of intermediate
  markings in path from Min to M

- Linear algebraic approach: vector addition systems:

  - For a finite net with n places, a marking is a vector
    (m1,m2,..,mn)

  - Each transtion is a change vector (delta1,delta2,..,deltam)

  - Effect of t at M is given by M+t.  Effect of a sequence only
    depends on the number of occurrences of each type of
    transition. 

  _ Represent a net by an initial vector Min and change vectors
    {t1,t2,...,tk}.   Reachability of M implies there exist
    j1,j2,...,jk such that

      M = Min + j1*t1 + j2*t2 + ... + jk*tk

  - This is sufficient by not necessary: a solution may exist but
    the ji's may not yield a feasible firing sequence ---
    intermediate marking may drop below 0 for some place

  - Nevertheless, original proof of decidability of reachability
    heavily uses linear algebra.  More recent proofs try to find
    alternative "simpler" arguments.

  - Algorithmic complexity of reachability 
    - EXPSPACE-hard (Lipton)
    - Non-primitive recursive upper bound (Leroux and Schmitz)
    - Nonelementary lower bound, 2018 (Czerwinski et al)

Coverability

- Compare markings component-wise: M <= M' if M(p) <= M'(p) for
  all p in P

- M' covers M if M' >= M

- Coverability question: Given M, does there exist M' in
  Reach(Min) such that M' covers M?

- Why is this useful?  Firing rule for nets is monotonic with
  respect to markings.

  If M--t--> and M' >= M, then M'--t--> as well.  In fact, a
  stronger statement holds:  If M--t-->M1 and M' >= M, then
  M'--t-->M1' and M1' >= M1.

- "Bad" markings are typically characterized in terms of the
  transitions they enable (e.g. in mutual exclusion, both
  access to critical section is simultaneously enabled in both
  processes).   If M is bad and M' >= M, M' is also bad.

- If we find M' in Reach(Min) that covers a bad marking M, we
  have an unsafe behaviour (though M may not be precisely
  reachable).  If no such M' is exists, it also implies M itself
  is unreachable, so the behaviour is safe.

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